Cup-type tappets for use in internal combustion engines are known in the art. They serve to provide automatic valve play compensation between the upper end of internal combustion engine valves and the camshaft. They comprise a piston/cylinder unit which is disposed in the loading direction of the respective valve and into which oil from the oil pressure circulating lubrication system is fed. This causes the piston/cylinder unit to extend to a length corresponding to the distance between the upper end of the closed engine valve and the camshaft when the cam is under no load. Leakage fluid leaves the piston/cylinder unit constantly in defined quantities through throttle gaps specially provided for this purpose. The brief loading of the cup-type tappet by the cam during the valve actuation is incapable of significantly changing the length of the piston/cylinder unit. Therefore, reliable closing of the valves is assured at all times independent of possibly occurring wear. However, in the prior art, the clatter noises occurring when starting the engine leaves much to be desired and can lead to engine damage.